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The log-dynamic brain: how skewed distributions affect network operations

Nature Reviews Neuroscience volume 15, pages 264–278 (2014)Cite this article

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Key Points

  • At many physiological and anatomical levels in the brain, the distribution of numerous parameters is strongly skewed with a heavy tail and typically follows a lognormal distribution.

  • The power and frequency relationship of brain oscillations is typically expressed in a log scale.

  • Network synchrony, measured as a fraction of spiking neurons in a given time window, shows lognormal distribution in all brain states.

  • Firing rates, spike bursts and synaptic weights follow a lognormal distribution. Importantly, these parameters remain correlated across brain states, environments and situations.

  • The log-dynamic patterns of networks may be supported by the lognormal distribution of corticocortical connections strengths and axon diameters.

  • A preconfigured, strongly connected minority of fast-firing neurons form the backbone of brain connectivity and serve as an ever-ready, fast-acting system. However, full performance of the brain also depends on the activity of very large numbers of weakly connected and slow-firing majority of neurons.

Abstract

We often assume that the variables of functional and structural brain parameters — such as synaptic weights, the firing rates of individual neurons, the synchronous discharge of neural populations, the number of synaptic contacts between neurons and the size of dendritic boutons — have a bell-shaped distribution. However, at many physiological and anatomical levels in the brain, the distribution of numerous parameters is in fact strongly skewed with a heavy tail, suggesting that skewed (typically lognormal) distributions are fundamental to structural and functional brain organization. This insight not only has implications for how we should collect and analyse data, it may also help us to understand how the different levels of skewed distributions — from synapses to cognition — are related to each other.

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Figure 1: Logarithmic distributions at macroscales.
Figure 2: Skewed distribution of the magnitude of population synchrony.
Figure 3: Lognormal distribution of firing rates in the cortex.
Figure 4: Firing rates of principal neurons are preserved across brain states and environments.
Figure 5: Lognormal distribution of synaptic weights and spike transfer probability.
Figure 6: Skewed distribution of axon calibres.

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Acknowledgements

The authors thank A. Berardino, S. Burke, T. Fukai, A. Grosmark, K. Harris, Y. Ikegaya, M. Kahana, C. Koch, J. Magee, A. Maurer, A. Peyrache, A. Reyes, E. Schomburg, L. Sjulson, S. Song, R. Tsien and S. Wang for comments and discussions. The authors are supported by the US National Institutes of Health (NS034994, MH54671 and NS074015 (to G.B.)), National Science Foundation (0542013), the J.D. McDonnell Foundation, Human Frontiers Science Program (grant RGP0032/2011 (to G.B.)), Uehara Memorial Foundation (K.M.), Astellas Foundation for Research on Metabolic Disorders (K.M.) and the Japan Society of Promotion for Sciences (K.M.).

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Authors and Affiliations

  1. New York University Neuroscience Institute, New York University Langone Medical Center, New York, 10016, New York, USA

    György Buzsáki & Kenji Mizuseki

  2. Center for Neural Science, New York University, New York, 10003, New York, USA

    György Buzsáki

  3. Allen Institute for Brain Science, 551 North 34th Street, Seattle, 98103, Washington, USA

    Kenji Mizuseki

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  1. György Buzsáki
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Correspondence to György Buzsáki.

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PowerPoint slides

Glossary

Spike transfer

The fraction of spikes in the postsynaptic neuron relative to the number of spikes in the presynaptic, driver neuron (or neurons). It is an indirect way of measuring synaptic strength.

Power laws

A term to describe a relationship between two variables, where one varies as a power of the other. The indication of a power law is a distribution of values on a straight line on a double log plot. The right tail of the lognormal distribution often follows a power law distribution.

Scale-free properties

Properties that characterize networks with a degree distribution that follows a power law, characterized by a heavy tail ('Pareto tail').

Cross-frequency phase–amplitude coupling

This is perhaps the most prominent 'law' underlying the hierarchy of the system of brain oscillators. The phase of the slower oscillation modulates the power of the faster rhythm (or rhythms).

Decibel

A logarithmic unit used to express the ratio between two values of a variable. It is often used to describe gain or attenuation: for example, the ratio of input and output.

Sharp-wave ripples

Patterns of activity in the hippocampus, consisting of a sharp wave reflecting the strong depolarization of the apical dendrites of pyramidal cells and a short-lived, fast oscillation ('ripple') as a result of the interaction between bursting pyramidal cells and perisomatic interneurons.

Theta oscillations

A prominent 4–10 Hz collective rhythm of the hippocampus. Other brain regions can also generate oscillations in this band.

Remap

This term refers to the observation that place cell representations can abruptly change.

Immediate-early gene

A gene that is rapidly and transiently activated in response to relevant stimuli.

Fos

A prominent immediate-early gene in the brain; it is often used as an indicator of neuronal activity.

Synaptic weights

A measure of the strength of the synapse, which determines the amplitude of the postsynaptic neuron's response to a presynaptic spike.

Up states

The active phases of the slow oscillation. Intracellularly, an up state corresponds to latching the membrane potential to a more depolarized, near-spiking threshold value.

Spike-timing-dependent plasticity

(STDP). A plasticity-inducing paradigm in which the relative timing of spikes between the pre- and postsynaptic neurons determines the direction and magnitude of the change in synaptic strength.

Wiring economy

The idea that connections among multiple brain regions and neurons are arranged to reduce energy cost and volume demand.

Feedforward inhibition

Excitatory afferents to the various domains of pyramidal cells are matched by parallel sets of inhibitory interneurons to filter, attenuate or route excitation. It can perform division operation.

Redundancy

This term refers to the observation that multiple replicas of input representations exist.

Degeneracy

In biology, this term refers to the idea that different solutions evolved to carry out the same functions.

Preconfigured brain

This term refers to an idea that connections and dynamics in the brain are largely self-generated and that experience matches the pre-existing patterns to the external world, thereby giving rise to 'meaning'.

Attractors

Hypothetical effectors that move elements of a system to more stable states over time. Inhibition-based brain rhythms often show properties of an attractor.

Internal models

A term derived from the hypothesis that the perceived world is not simply a reflection of the objective reality but depends on previous experience and brain state. In this hypothesis, internal models reflect the source of our individual views.

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Buzsáki, G., Mizuseki, K. The log-dynamic brain: how skewed distributions affect network operations. Nat Rev Neurosci 15, 264–278 (2014). https://doi.org/10.1038/nrn3687

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